Estimating the effective lifetime of the $z\sim6$ quasar population from the composite proximity zone profile

The lifetime of quasars can be estimated by means of their proximity zone sizes, which are regions of enhanced flux bluewards of the Lyman-$\alpha$ emission line observed in the rest-frame UV spectra of high-redshift quasars, because the intergalactic gas has a finite response time to the quasars' radiation. We estimate the effective lifetime of the high-redshift quasar population from the composite transmitted flux profile within the proximity zone region of a sample of $15$ quasars at $5.8\leq z\leq 6.6$ with precise systemic redshifts, and similar luminosities, i.e. $-27.6\leq M_{1450}\leq-26.4$, and thus a similar instantaneous ionizing power. We develop a Bayesian method to infer the effective lifetime from the composite spectrum, including robust estimates of various sources of uncertainty on the spectrum. We estimate an effective lifetime of the quasar population as a whole of $\log_{10}(t_{Q}/{yr}) = 5.7^{+0.5 (+0.8)}_{-0.3 (-0.5)}$ given by the median and $68$th ($95$th) percentile of the posterior probability distribution. While our result is consistent with previous quasar lifetime studies, it poses significant challenges on the current model for the growth of supermassive black holes (SMBHs) located in the center of the quasars' host galaxies, which requires that quasar lifetimes are more than an order of magnitude longer.